Analytical Study of a Kerr-Sen Black Hole and a Charged Massive Scalar Field

It is reported that Kerr-Newman and Kerr-Sen black holes are unstable to perturbations of charged massive scalar field. In this paper, we study analytically the complex frequencies which characterize charged massive scalar fields in a near-extremal Kerr-Sen black hole. For near-extremal Kerr-Sen black holes and for charged massive scalar fields in the eikonal large-mass $\mathcal{M}\gg \mu$ regime, where $\mathcal{M}$ is the mass of the black hole, and $\mu$ is the mass of the charged scalar field, we have obtained a simple expression for the dimensionless ratio $\omega_I /\left(\omega_R - \omega_c\right)$, where $\omega_I$ and $\omega_R$ are, respectively, the imaginary and real parts of the frequency of the modes, and $\omega_c$ is the critical frequency for the onset of super-radiance. We have also found our expression is consistent with the result of Hod [Phys. Rev. D 94, 044036 (2016)] for the case of a near-extremal Kerr-Newman black hole and the result of Zouros and Eardly [Ann. Phys. (N.Y.) 118, 139 (1979)] for the case of neutral scalar fields in the background of a near-extremal Kerr black hole.